Packaging of tetrahedra is the task of arranging identical regular tetrahedra in three-dimensional space so as to fill as much of the space as possible.
At present, the best limit of the packing density obtained for the optimal packing of regular tetrahedra is the number of 85.63% [1] . Tetrahedra do not pave the space [2] and, as is known, the upper filling limit is below 100% (namely, 1 - (2.6…) · 10 −25 ) [3] .
Content
Historical Results
Aristotle argued that tetrahedra should fill the space completely [4] .
In 2006, Conway and Torquato showed that a packing density of about 72% can be obtained by constructing a tetrahedron lattice that is not a Bravais lattice (with several parts having different orientations), and showed that the best packaging of tetrahedra cannot be a lattice packing (with one element on a repeating block and when each element has the same orientation) [5] . These constructions nearly double the optimal packing density on the basis of the Bravae lattice, which Hoyleman received and whose density is 36.73% [6] . In 2007 and 2010, Chaykin and colleagues showed that bodies similar to a tetrahedron can be randomly packed into a final container with a packing density of between 75% and 76% [7] . In 2008, Chen first proposed the packaging of regular tetrahedra, which is denser than the packaging of spheres, namely, 77.86% [8] [9] . Improvements made Torquato and Jiao in 2009, squeezing Chen’s design using a computer algorithm and obtaining a packing share of 78.2021% [10] .
In mid-2009, Haji-Akbari and co-authors showed, using the Monte-Carlo method for an initially random system with a packing density of> 50%, that the equilibrium flux of solid tetrahedra spontaneously transforms into a twelve-angle quasicrystal that can be compressed to 83.24%. They also described chaotic packaging with a density greater than 78%. For periodic approximation by quasicrystals with a cell of 82 tetrahedra, they obtained a packing density of 85.03% [11] .
At the end of 2009, a new, simpler package family with a density of 85.47% was discovered by Callus, Elzer and Gravel [12] . On the basis of these packages, by slightly improving them, Torquato and Jiao obtained a density of 85.55% at the end of 2009 [13] . In early 2010, Chen, Engel and Glotzer obtained a density of 85.63% [1] , and now this result is the densest packing of regular tetrahedra.
Linking to other packaging tasks
Since the early known boundaries of the packing density of tetrahedra were smaller than the packing of balls , it was suggested that a regular tetrahedron could be a counterexample of that the optimal packing density of identical balls is less than the packing density of any other body. More recent studies have shown that this is not the case.
See also
- Packaging tasks
- - packaging of irregular tetrahedra in 3-dimensional space.
- is a packaging based on regular tetrahedra.
Notes
- ↑ 1 2 Chen, Engel, Glotzer, 2010 , p. 253–280.
- ↑ Struik, 1925 , p. 121-134.
- ↑ Gravel, Elser, Kallus, 2010 , p. 799–818.
- ↑ Polster, Ross, 2011 .
- ↑ Conway, 2006 , p. 10612–10617.
- ↑ Hoylman, 1970 , p. 135–138.
- ↑ Jaoshvili, Esakia, Porrati, Chaikin, 2010 , p. 185501.
- ↑ Chen, 2008 , p. 214–240.
- ↑ Cohn, 2009 , p. 801–802.
- ↑ Torquato, Jiao, 2009 , p. 876–879.
- ↑ Haji-Akbari, Engel, Keys, Zheng et al., 2009 , p. 773–777.
- ↑ Kallus, Elser, Gravel, 2010 , p. 245-252.
- ↑ Torquato, Jiao, 2009 .
Literature
- Elizabeth R. Chen, Michael Engel, Sharon C. Glotzer. Dense crystalline dimer packings of regular tetrahedra // Discrete & Computational Geometry . - 2010. - Vol. 44 , no. 2 - p . 253–280 . - DOI : 10.1007 / s00454-010-9273-0 .
- DJ Struik. De impletione loci // Nieuw Arch. Wiskd. . - 1925. - T. 15 . - P. 121–134 .
- Simon Gravel, Veit Elser, Yoav Kallus. Upper boundaries of the tetrahedra and octahedral // Discrete & Computational Geometry . - 2010. - Vol . 46 . - p . 799–818 . - DOI : 10.1007 / s00454-010-9304-x . - arXiv : 1008.2830 .
- JH Conway. Packing, tiling, and covering with tetrahedral // Proceedings of the National Academy of Sciences . - 2006. - Vol. 103 , no. 28 - p . 10612–10617 . - DOI : 10.1073 / pnas.0601389103 . - . - PMID 16818891 .
- Douglas J. Hoylman. The densest lattice packing of tetrahedral // Bulletin of the American Mathematical Society . - 1970. - V. 76 . - pp . 135–138 . - DOI : 10.1090 / S0002-9904-1970-12400-4 .
- Alexander Jaoshvili, Andria Esakia, Massimo Porrati, Paul M. Chaikin. Experiments on the Random Packing of Tetrahedral Dice // Physical Review Letters . - 2010. - Vol. 104 , no. 18 - p . 185501 . - DOI : 10.1103 / PhysRevLett.104.185501 . - . - PMID 20482187 .
- Elizabeth R. Chen. A Dense Packing of Regular Tetrahedra // Discrete & Computational Geometry . - 2008. - V. 40 , no. 2 - p . 214–240 . - DOI : 10.1007 / s00454-008-9101-y .
- Henry Cohn. Mathematical physics: A tight squeeze // Nature . - 2009. - V. 460 , no. 7257 . - p . 801–802 . - DOI : 10.1038 / 460801a . - . - PMID 19675632 .
- S. Torquato, Y. Jiao. Dense packings of the Platonic and Archimedean solids // Nature . - 2009. - V. 460 , no. 7257 . - p . 876–879 . - DOI : 10.1038 / nature08239 . - . - arXiv : 0908.4107 . - PMID 19675649 .
- Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray, Sharon C. Glotzer. Disordered, quasicrystalline and crystalline phases of densely packed tetrahedral // Nature . - 2009. - V. 462 , no. 7274 . - p . 773–777 . - DOI : 10.1038 / nature08641 . - . - arXiv : 1012.5138 . - PMID 20010683 .
- Yoav Kallus, Veit Elser, Simon Gravel. Dense Periodic Packings of Tetrahedra with Small Repeating Units // Discrete & Computational Geometry . - 2010. - T. 44 . - P. 245–252. - DOI : 10.1007 / s00454-010-9254-3 .
- Torquato, S. & Jiao, Y. (2009), "Analytical Constructions of the Family of Det Tetrahedron Packings and the Role of Symmetry", arΧiv : 0912.4210 [cond-mat.stat-mech]
- Burkard Polster and Marty Ross . Do women have fewer teeth than men? (March 14, 2011).
Links
- Packing Tetrahedrons, Perfect Fit , NYTimes
- Efficient shapes , The Economist
- Pyramids are the best shape for packing , New Scientist